§10.prop3.109ἀπὸ μέσου ῥητοῦ ἀφαιρουμένου ἄλλαι δύο ἄλογοι γίνονται ἤτοι μέσης ἀποτομὴ πρώτη ἢ μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσα.
If from a medial area a rational area be subtracted, two other irrational straight lines arise, either a first apotome of a medial or that which produces with a rational area a medial whole.
ἀπὸ γὰρ μέσου τοῦ ΒΓ ῥητὸν ἀφῃρήσθω τὸ ΒΔ. λέγω, ὅτι ἡ τὸ λοιπὸν τὸ ΕΓ δυναμένη μία δύο ἀλόγων γίνεται ἤτοι μέσης ἀποτομὴ πρώτη ἢ μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσα. Ἐκκείσθω γὰρ ῥητὴ ἡ ΖΗ, καὶ παραβεβλήσθω ὁμοίως τὰ χωρία.
For from the medial area BC let the rational area BD be subtracted; I say that the straight line producing the remaining area EG is one of two irrational straight lines, either a first apotome of a medial or that which produces with a rational area a medial whole. For from the medial area BC let the rational area BD be subtracted; I say that the straight line producing the remaining area EG is one of two irrational straight lines, either a first apotome of a medial or that which produces with a rational area a medial whole. For let the rational straight line ZH be set out, and let the areas be applied in the same way.
ἔστι δὴ ἀκολούθως ῥητὴ μὲν ἡ ΖΘ καὶ ἀσύμμετρος τῇ ΖΗ μήκει, ῥητὴ δὲ ἡ ΚΖ καὶ σύμμετρος τῇ ΖΗ μήκει· αἱ ΖΘ, ΖΚ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΚΘ, προσαρμόζουσα δὲ ταύτῃ ἡ ΖΚ. ἤτοι δὴ ἡ ΘΖ τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ ἢ τῷ ἀπὸ ἀσυμμέτρου.
Consequently, ZΘ is rational and incommensurable in length with ZH, and KZ is rational and commensurable in length with ZH; therefore ZΘ, ZK are rational straight lines commensurable in square only; therefore KΘ is an apotome, and ZK is the annex to it. Either then ΘZ is greater in square than ZK by the square on a straight line commensurable with itself, or by the square on a straight line incommensurable with itself. therefore ZΘ, ZK are rational straight lines commensurable in square only; therefore KΘ is an apotome, and ZK is the annex to it. Either then ΘZ is greater in square than ZK by the square on a straight line commensurable with itself, or by the square on a straight line incommensurable with itself.
εἰ μὲν οὖν ἡ ΘΖ τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καί ἐστιν ἡ προσαρμόζουσα ἡ ΖΚ σύμμετρος τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΖΗ, ἀποτομὴ δευτέρα ἐστὶν ἡ ΚΘ. ῥητὴ δὲ ἡ ΖΗ· ὥστε ἡ τὸ ΛΘ, τουτέστι τὸ ΕΓ, δυναμένη μέσης ἀποτομὴ πρώτη ἐστίν. εἰ δὲ ἡ ΘΖ τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου, καί ἐστιν ἡ προσαρμόζουσα ἡ ΖΚ σύμμετρος τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΖΗ, ἀποτομὴ πέμπτη ἐστὶν ἡ ΚΘ· ὥστε ἡ τὸ ΕΓ δυναμένη μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσά ἐστιν· ὅπερ ἔδει δεῖξαι.
If then ΘZ is greater in square than ZK by the square on a straight line commensurable with itself, and the annex ZK is commensurable in length with the rational straight line ZH set out, KΘ is a second apotome. And ZH is rational; so that the straight line producing LΘ, that is, EG, is a first apotome of a medial. If then ΘZ is greater in square than ZK by the square on a straight line commensurable with itself, and the annex ZK is commensurable in length with the rational straight line ZH set out, KΘ is a second apotome. so that the straight line producing LΘ, that is, EG, is a first apotome of a medial. But if ΘZ is greater in square than ZK by the square on a straight line incommensurable with itself, and the annex ZK is commensurable in length with the rational straight line ZH set out, KΘ is a fifth apotome; so that the straight line producing EG is that which produces with a rational area a medial whole; which was to be proved. But if ΘZ is greater in square than ZK by the square on a straight line incommensurable with itself, and the annex ZK is commensurable in length with the rational straight line ZH set out, KΘ is a fifth apotome; so that the straight line producing EG is that which produces with a rational area a medial whole; which was to be proved.
§10.prop3.110ἀπὸ μέσου μέσου ἀφαιρουμένου ἀσυμμέτρου τῷ ὅλῳ αἱ λοιπαὶ δύο ἄλογοι γίνονται ἤτοι μέσης ἀποτομὴ δευτέρα ἢ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα. ἀφῃρήσθω γὰρ ὡς ἐπὶ τῶν προκειμένων καταγραφῶν ἀπὸ μέσου τοῦ ΒΓ μέσον τὸ ΒΔ ἀσύμμετρον τῷ ὅλῳ· λέγω, ὅτι ἡ τὸ ΕΓ δυναμένη μία ἐστὶ δύο ἀλόγων ἤτοι μέσης ἀποτομὴ δευτέρα ἢ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα. ἐπεὶ γὰρ μέσον ἐστὶν ἑκάτερον τῶν ΒΓ, ΒΔ, καὶ ἀσύμμετρον τὸ ΒΓ τῷ ΒΔ, ἔσται ἀκολούθως ῥητὴ ἑκατέρα τῶν ΖΘ, ΖΚ καὶ ἀσύμμετρος τῇ ΖΗ μήκει. καὶ ἐπεὶ ἀσύμμετρόν ἐστι τὸ ΒΓ τῷ ΒΔ, τουτέστι τὸ ΗΘ τῷ ΗΚ, ἀσύμμετρος καὶ ἡ ΘΖ τῇ ΖΚ· αἱ ΖΘ, ΖΚ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΚΘ.
If from a medial area there be subtracted a medial area which is incommensurable with the whole, the remaining two irrational straight lines arise, either a second apotome of a medial or that which produces with a medial area a medial whole. For, as in the preceding diagrams, let there be subtracted from the medial area BC the medial area BD incommensurable with the whole; I say that the straight line producing EG is one of two irrational straight lines, either a second apotome of a medial or that which produces with a medial area a medial whole. For, as in the preceding diagrams, let there be subtracted from the medial area BC the medial area BD incommensurable with the whole; I say that the straight line producing EG is one of two irrational straight lines, either a second apotome of a medial or that which produces with a medial area a medial whole. For since each of BC, BD is medial, and BC is incommensurable with BD, consequently each of ZΘ, ZK is rational and incommensurable in length with ZH. And since BC is incommensurable with BD, that is, HΘ with HK, ΘZ is also incommensurable with ZK; therefore ZΘ, ZK are rational straight lines commensurable in square only; therefore KΘ is an apotome. For since each of BC, BD is medial, and BC is incommensurable with BD, consequently each of ZΘ, ZK is rational and incommensurable in length with ZH. And since BC is incommensurable with BD, that is, HΘ with HK, ΘZ is also incommensurable with ZK; therefore ZΘ, ZK are rational straight lines commensurable in square only; therefore KΘ is an apotome.
εἰ μὲν δὴ ἡ ΖΘ τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καὶ οὐθετέρα τῶν ΖΘ, ΖΚ σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΖΗ, ἀποτομὴ τρίτη ἐστὶν ἡ ΚΘ. ῥητὴ δὲ ἡ ΚΛ, τὸ δʼ ὑπὸ ῥητῆς καὶ ἀποτομῆς τρίτης περιεχόμενον ὀρθογώνιον ἄλογόν ἐστιν, καὶ ἡ δυναμένη αὐτὸ ἄλογός ἐστιν, καλεῖται δὲ μέσης ἀποτομὴ δευτέρα· ὥστε ἡ τὸ ΛΘ, τουτέστι τὸ ΕΓ, δυναμένη μέσης ἀποτομή ἐστι δευτέρα.
If then ZΘ is greater in square than ZK by the square on a straight line commensurable with itself, and neither of ZΘ, ZK is commensurable in length with the rational straight line ZH set out, KΘ is a third apotome. If then ZΘ is greater in square than ZK by the square on a straight line commensurable with itself, and neither of ZΘ, ZK is commensurable in length with the rational straight line ZH set out, KΘ is a third apotome. And KL is rational; but the rectangle contained by a rational straight line and a third apotome is irrational, and the straight line producing it is irrational, and is called a second apotome of a medial; so that the straight line producing LΘ, that is, EG, is a second apotome of a medial.
εἰ δὲ ἡ ΖΘ τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ, καὶ οὐθετέρα τῶν ΘΖ, ΖΚ σύμμετρός ἐστι τῇ ΖΗ μήκει, ἀποτομὴ ἕκτη ἐστὶν ἡ ΚΘ. τὸ δʼ ὑπὸ ῥητῆς καὶ ἀποτομῆς ἕκτης ἡ δυναμένη ἐστὶ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα.
But if ZΘ is greater in square than ZK by the square on a straight line incommensurable with itself, and neither of ΘZ, ZK is commensurable in length with ZH, KΘ is a sixth apotome. But the straight line producing the area contained by a rational straight line and a sixth apotome is that which produces with a medial area a medial whole.
ἡ τὸ ΛΘ ἄρα, τουτέστι τὸ ΕΓ, δυναμένη μετὰ μέσου μέσον τὸ ὅλον ποιοῦσά ἐστιν· ὅπερ ἔδει δεῖξαι.
Therefore the straight line producing LΘ, that is, EG, is that which produces with a medial area a medial whole; which was to be proved.